Let's solve the system of linear equations to find the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]. The given system is:
[tex]\[
\begin{cases}
3a + 6b = 45 \\
2a - 2b = -12
\end{cases}
\][/tex]
Step 1: Simplify the equations if possible
Let's look at the second equation:
[tex]\[
2a - 2b = -12
\][/tex]
We can divide every term by 2 to simplify:
[tex]\[
a - b = -6 \implies a = b - 6
\][/tex]
Now we have:
[tex]\[
3a + 6b = 45
\][/tex]
Step 2: Substitute [tex]\( a \)[/tex] in the first equation
From the simplified second equation [tex]\( a = b - 6 \)[/tex], we substitute [tex]\( a \)[/tex] into the first equation:
[tex]\[
3(b - 6) + 6b = 45
\][/tex]
Expand and simplify:
[tex]\[
3b - 18 + 6b = 45
\][/tex]
Combine like terms:
[tex]\[
9b - 18 = 45
\][/tex]
Add 18 to both sides:
[tex]\[
9b = 63
\][/tex]
Divide by 9:
[tex]\[
b = 7
\][/tex]
Step 3: Find [tex]\( a \)[/tex]
Now that we have [tex]\( b \)[/tex], substitute [tex]\( b \)[/tex] back into [tex]\( a = b - 6 \)[/tex]:
[tex]\[
a = 7 - 6 = 1
\][/tex]
Thus, the solution to the system is [tex]\( (a, b) = (1, 7) \)[/tex].
Step 4: Verify the solution
To ensure that our solution is correct, substitute [tex]\( a = 1 \)[/tex] and [tex]\( b = 7 \)[/tex] back into the original equations:
First equation:
[tex]\[
3(1) + 6(7) = 3 + 42 = 45 \quad \text{(Correct)}
\][/tex]
Second equation:
[tex]\[
2(1) - 2(7) = 2 - 14 = -12 \quad \text{(Correct)}
\][/tex]
So, the solution is indeed correct.
The correct solution [tex]\( (a, b) \)[/tex] for the given system of linear equations is:
[tex]\[
(1, 7)
\][/tex]
Among the given choices:
- [tex]\((-27, 6)\)[/tex]
- [tex]\((-1, 7)\)[/tex]
- [tex]\((1, 7)\)[/tex]
- [tex]\((27, -6)\)[/tex]
The solution is [tex]\( (1, 7) \)[/tex].
